75.20.24 problem 663
Internal
problem
ID
[17087]
Book
:
A
book
of
problems
in
ordinary
differential
equations.
M.L.
KRASNOV,
A.L.
KISELYOV,
G.I.
MARKARENKO.
MIR,
MOSCOW.
1983
Section
:
Chapter
2
(Higher
order
ODEs).
Section
15.5
Linear
equations
with
variable
coefficients.
The
Lagrange
method.
Exercises
page
148
Problem
number
:
663
Date
solved
:
Monday, March 31, 2025 at 03:40:23 PM
CAS
classification
:
[[_2nd_order, _missing_y]]
\begin{align*} y^{\prime \prime }-2 \tan \left (x \right ) y^{\prime }&=1 \end{align*}
✓ Maple. Time used: 0.002 (sec). Leaf size: 31
ode:=diff(diff(y(x),x),x)-2*tan(x)*diff(y(x),x) = 1;
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {\ln \left (1+\cos \left (2 x \right )\right )}{4}+\frac {\ln \left (\cos \left (x \right )\right )}{2}+\frac {\left (2 x +4 c_1 \right ) \tan \left (x \right )}{4}+c_2
\]
✓ Mathematica. Time used: 0.056 (sec). Leaf size: 37
ode=D[y[x],{x,2}]-2*Tan[x]*D[y[x],x]==1;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \int _1^x\sec ^2(K[2]) \left (c_1+\int _1^{K[2]}\cos ^2(K[1])dK[1]\right )dK[2]+c_2
\]
✓ Sympy. Time used: 5.154 (sec). Leaf size: 170
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-2*tan(x)*Derivative(y(x), x) + Derivative(y(x), (x, 2)) - 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = C_{1} + \frac {C_{2} \sin {\left (x \right )}}{\cos {\left (x \right )}} - \frac {x \tan {\left (\frac {x}{2} \right )}}{\tan ^{2}{\left (\frac {x}{2} \right )} - 1} - \frac {\log {\left (\cos {\left (x \right )} \right )}}{2} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )} + \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )} - \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{2}{\left (\frac {x}{2} \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )} + \frac {\log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{2 \left (\tan ^{2}{\left (\frac {x}{2} \right )} - 1\right )}
\]